Convolution & LTI systems worksheet with answers

Free printable Convolution & LTI systems worksheet: 6 problems from easy to exam level, with an answer key. Print it or save it as a PDF.

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1.Easy
x[n]n−1011−22434h[n]n−10−31−32−134

The sequences x[n] = {1, −2, 4} and h[n] = {−3, −3, −1} both start at n = 0. Find the largest sample ymax of y[n] = x[n] ∗ h[n].

x[n] = {1, −2, 4}, n = 0…2h[n] = {−3, −3, −1}, n = 0…2
ymax =
2.Easy
x[n]n−101323314h[n]n−104132

The sequence x[n] is nonzero for n = 1…3 and h[n] for n = 0…1, as shown. Find the last index nlast at which y[n] = x[n] ∗ h[n] can be nonzero.

x[n]: n = 1…3h[n]: n = 0…1
nlast =
3.Medium
x(t)t32h(t)t41

The pulse x(t) = 2 for 0 ≤ t < 3 s (zero elsewhere) is applied to a system with the impulse response h(t) = 1 for 0 ≤ t < 4 s (zero elsewhere). Find the output y(5.5) at t = 5.5 s.

x(t) = 2, 0 ≤ t < 3 sh(t) = 1, 0 ≤ t < 4 st = 5.5 s
y(5.5) =
4.Medium
x(t)t210h(t)t24

The pulse x(t) = 10 for 0 ≤ t < 2 s is convolved with h(t), a triangle that falls linearly from 4 at t = 0 to 0 at t = 2 s. Find the area Ay under y(t) = x(t) ∗ h(t).

x(t) = 10, 0 ≤ t < 2 sh(0) = 4, h(2) = 0
Ay =
5.Hard
x(t)t (ms)5102010−10

A triangle wave rises and falls linearly between −10 and +10 with period T = 10 ms. Find the amplitude c5 of its Fourier component at 500 Hz.

±10T = 10 msf = 500 Hz
c5 =
6.Hard
δ(t)y′ + 1y = 10xh(t)

A system obeys y′(t) + 1·y(t) = 10·x(t). Find its impulse response at t = 1 s, h(1).

y′ + 1y = 10xt = 1 s
h(1) =

Answer key

  1. 1. ymax = 3
  2. 2. nlast = 4
  3. 3. y(5.5) = 3
  4. 4. Ay = 80
  5. 5. c5 = 0.3242
  6. 6. h(1) = 3.679

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